Queues with Server Vacations and Levy Processes with Secondary Jump Input

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Abstract
Motivated by models of queues with server vacations, we consider a Le ´ vy process modified to have random,jumps,at arbitrary stopping times. The extra jumps can counteract a drift in the Le´ vy process so that the overall Le ´ vy process with secondary jump input, can have a proper limiting distribution. For example, the workload process in an M/G/1 queue with a server vacation each time the server finds an empty system is such a Le ´ vy process with secondaryjump input. We show,that a certain functional of a Le ´ vy process,with secondary,jump,input is a martingale and we apply,this martingale to characterize the steady-state distribution. We establish stochastic decomposition,results for the case in which the Le ´ vy process has no negative jumps, which extend and unify previous decomposition results for the workload process in the M/G/1 queue with,server vacations and Brownian motion with secondary,jump input. We also apply martingales to provide a new,proof of the,known,simple,form,of the steady-state distribution of the associated reflected Le ´ vy process when,the Le ´ vy process,has no negative jumps (the generalized Pollaczek-Khinchine formula).

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